Question

Dan McClure is trying to decide on how many copies of a book to purchase at the start of the upcoming selling season for his bookstore. The book retails at $28.00. The publisher sells the book to Dan for $20.00. Dan will dispose of all of the unsold copies of the book at 75 percent off the retail price, at the end of the season. Dan estimates that demand for this book during the season is normal with a mean of 100 and a standard deviation of 42.

How many copies should Dan order to maximize his expected profit?

The publisher’s variable cost per book is $7.50. Given the order quantity in part a, what is the

publisher’s expected profit?

The publisher is thinking of offering the following deal to Dan. At the end of the season, the publisher will buy back unsold copies at a predetermined price of $15.00. However, Dan would have to bear the costs of shipping unsold copies back to the publisher at $1.00 per copy.

c. How many books should Dan order to maximize his expected profits given the buy-back offer?
d. Assume the publisher is able on average to earn $6 on each returned book net the publisher’s handling

costs (some books are destroyed while others are sold at a discount and others are sold at full price). Suppose the publisher continues to charge $20 per book and Dan still incurs a $1 cost to ship each book back to the publisher. What price should the publisher pay Dan for returned books to maximize the supply chain’s profit (the sum of the publisher’s profit and Dan’s profit)?

Answer 1

Part A: Order Quantity to Maximize Dan's Expected Profit

Optimal Order Quantity Q:

`\[ Q = 100 + Z \cdot 42 \]`

`(Where \(Z\) is the Z-score corresponding to the desired service level)`

Part B: Publisher's Expected Profit

Publisher's Expected Profit:

`\[ \text{Publisher's Profit} = (\$28.00 - \$20.00 - \$7.50) \cdot \text{Expected Demand} \]`

Part C: Buy-Back Offer

Optimal Order Quantity with Buy-Back Offer
`(\(Q_{\text{buyback}}\))`:

`\[ Q_{\text{buyback}} = 100 + Z \cdot 42 \]`

(Using the same formula as Part A)

Total Cost to Dan:

`\[ \text{Total Cost to Dan} = Q_{\text{buyback}} \cdot (\$20.00 + \$1.00) \]`

Part D: Optimal Buy-Back Price for Maximum Supply Chain Profit

Supply Chain Profit
`(\(P_{\text{supply chain}}\))`:

`\[ P_{\text{supply chain}} = \text{Dan's Profit} + \text{Publisher's Profit} \]`

Optimal Buy-Back Price:

`\[ \text{Optimal Buy-Back Price} = \$20.00 + \text{Publisher's Average Profit on Returns} \]`

Step-by-step explanation:

Part A.

The optimal order quantity considers the trade-off between lost sales and holding costs. By using the normal distribution and the desired service level, Dan can determine the order quantity that maximizes expected profit.

Paet B.

The publisher's profit is the difference between the selling price to Dan and the variable cost per book, multiplied by the expected demand.

Part C.

The buy-back offer introduces the cost of shipping unsold copies back to the publisher, affecting the total cost to Dan.

Part D.

Maximizing the supply chain profit involves finding the buy-back price that balances Dan's and the publisher's profits.